A LevelMathsOCRMaths APureExam QuestionsIntegrationIntegration

Integration (OCR A Level Maths A: Pure): Exam Questions

Exam code: H240

3 hours36 questions
Easy
Medium
Hard
Very Hard
1
3 marks

Find

(i) integral 2 x space straight d x

(ii) integral 6 x squared space straight d x

(iii) integral 1 half x to the power of 1 half end exponent space straight d x

How did you do?

2
4 marks

Find the value of

(i) integral subscript 1 superscript 2 4 x space straight d x

(ii) integral subscript 0 superscript 3 open parentheses 9 x squared plus 4 x close parentheses space straight d x

How did you do?

3
2 marks

Find

integral open parentheses 3 x squared plus 5 x plus 3 close parentheses space straight d x

writing each term in simplest form.

How did you do?

4
2 marks

Find

integral open parentheses 3 x to the power of 1 half end exponent plus 2 x to the power of negative 1 half space end exponent close parentheses straight d x

writing your answer in simplest form.

How did you do?

5
2 marks

Find the value of

integral subscript 1 superscript 5 left parenthesis 4 x plus 6 x squared right parenthesis space straight d x

How did you do?

6a
2 marks

Write

fraction numerator 3 x cubed plus 4 x to the power of 6 over denominator x squared end fraction

in the form  3 x to the power of a plus 4 x to the power of b, where a and b are constants to be found.

How did you do?

6b
2 marks

Hence find

integral open parentheses fraction numerator 3 x cubed plus 4 x to the power of 6 over denominator x squared end fraction close parentheses space straight d x

writing your answer in simplest form.

How did you do?

7a
2 marks

Find

integral open parentheses 5 x to the power of 4 plus 6 x squared plus 2 x plus 3 close parentheses space straight d x

How did you do?

7b
2 marks

Given that

  • space straight f to the power of apostrophe left parenthesis x right parenthesis equals 5 x to the power of 4 plus 6 x squared plus 2 x plus 3

  • the curve space y equals straight f left parenthesis x right parenthesis passes through the point open parentheses 1 comma space 10 close parentheses

find straight f left parenthesis x right parenthesis, giving your answer in simplest form.

How did you do?

8
4 marks

A curve, C, is defined by the following:

  • fraction numerator straight d y over denominator straight d x end fraction equals 2 x cubed minus x

  • C passes through the point open parentheses 2 comma space minus 2 close parentheses

Show that the equation of C can be written in the form

space 2 y equals x to the power of 4 minus x squared plus k

where k is a constant to be found.

How did you do?

9a
1 mark

The area bounded by the curve with equation space y equals 9 minus x squared, the x-axis and the vertical lines with equations x equals 1 and x equals 2 is shaded below.

Graph of y = 9 - x² with a shaded area between x = 1 and x = 2 and the x axis, bounded by dashed lines.

Write down a definite integral that represents this area.

How did you do?

9b
2 marks

Use algebraic integration to find the exact area of the shaded region in part (a).

How did you do?

10
4 marks

The diagram below shows the curve with equation y equals 7 x minus x squared minus 6 passing through the points open parentheses 1 comma space 0 close parentheses and open parentheses 6 comma space 0 close parentheses on the x-axis.

q9-8-1-integration-easy-a-level-maths-pure-screenshot

Use algebraic integration to find the exact area of the shaded region bounded by the curve and the x-axis.

How did you do?

11
4 marks

The diagram below shows the curve with equation y equals x squared minus 8 x plus 12 passing through the points open parentheses 2 comma space 0 close parentheses and open parentheses 6 comma space 0 close parentheses on the x-axis.

q10-8-1-integration-easy-a-level-maths-pure-screenshot

Use algebraic integration to find the exact area of the shaded region.

How did you do?

12
4 marks

A curve C has the equation y equals straight f open parentheses x close parentheses

Given that

  • straight f apostrophe open parentheses x close parentheses equals 3 minus 2 x plus 6 x squared

  • the curve passes through the point open parentheses 4 comma space 64 close parentheses

find the equation of the curve, giving your answer in simplest form.

How did you do?

13a
1 mark

Simplify 

space x minus open parentheses x squared minus 8 x plus 18 close parentheses

How did you do?

13b
4 marks

The diagram below shows the curve with equation y equals x squared minus 8 x plus 18 intersecting the straight line y equals x at the points open parentheses 3 comma space 3 close parentheses and open parentheses 6 comma space 6 close parentheses.

q11-8-1-integration-easy-a-level-maths-pure-screenshot

Use algebraic integration to find the exact area of the shaded region.

How did you do?

14
4 marks

The finite region R, shown in the figure below, is bounded by the curve with equation y equals x squared minus 4 x plus 3 and the x-axis.

Use algebraic integration to find the exact area of R.

q11-8-1-integration-medium-a-level-maths-pure-screenshot

How did you do?

15
4 marks

The figure below shows the finite shaded region, R, bounded by the curve y equals 2 x plus 3 x squared minus 2 x cubed and the x-axis.

q5-8-1-integration-hard-a-level-maths-pure-screenshot

Two x-intercepts of y equals 2 x plus 3 x squared minus 2 x cubed are shown, 0 and 2.

Use algebraic integration to find the exact area of R.

How did you do?

16
Sme Calculator
3 marks

Find

integral open parentheses 3 minus 2 x close parentheses squared space straight d x

writing each term in simplest form.

How did you do?

1
4 marks

Find the value of the positive constant k such that

integral subscript k superscript 5 space space left parenthesis 2 x minus 1 right parenthesis space straight d x equals 20

How did you do?

2a
3 marks

Find

integral open parentheses x squared plus 5 minus 3 over x squared close parentheses space straight d x

giving your answer in simplest form.

How did you do?

2b
2 marks

Write down

integral 3 straight e to the power of 3 x end exponent space d x

How did you do?

3
6 marks

The figure below shows a sketch of the line y equals 2 and the curve with equation y equals x squared minus 4 x plus 5.

q9-8-1-integration-medium-a-level-maths-pure-screenshot

Use algebraic integration to find the exact area of R.

How did you do?

4
7 marks

The curve with equation y equals x squared and the straight line with equation y equals 6 minus x are shown in the figure below.

The finite region bounded by the curve, the line and the x-axis is shaded.

Graph showing curves y = x^2 and y = 6-x intersecting at point Q. The area in the first quadrant enclosed by the curve, the line and the x-axis is shaded. The x-intercept of the curve is labelled P, the x-intercept of the line is labelled R, and the point of intersection between the curve and line in the first quadrant is labelled Q.

Use algebraic integration to find the exact area of the shaded region.

How did you do?

5
5 marks

Use algebraic integration to find the exact value of

integral subscript 4 superscript 9 fraction numerator x squared plus 1 over denominator square root of x end fraction space straight d x

How did you do?

6
5 marks

The finite region R, shown in the figure below, is bounded by the curve with equationspace y equals 4 minus open parentheses x minus 2 close parentheses squared and the x-axis.

q5-8-1-integration-easy-a-level-maths-pure-screenshot

The x-intercepts of the curve space y equals 4 minus open parentheses x minus 2 close parentheses squared are 0 and 4.

Use algebraic integration to find the exact area of R.

How did you do?

7
3 marks

Find

integral open parentheses 2 square root of x plus 5 x to the power of 1 third end exponent close parentheses space straight d x

giving your answer in simplest form.

How did you do?

8
4 marks

A curve has the equation y equals straight f open parentheses x close parentheses

Given that

  • straight f to the power of apostrophe open parentheses x close parentheses equals 3 minus 2 x plus 4 over x squared

  • the curve passes through the point open parentheses negative 2 comma space 3 close parentheses

find the equation of the curve, giving your answer in simplest form.

How did you do?

9
3 marks

Find the integer value of b such that

integral subscript negative 3 end subscript superscript 4 open parentheses 2 k x plus 3 k x squared close parentheses space straight d x equals b k

where k is a constant.

How did you do?

10a
3 marks

A curve has the equation y equals straight f open parentheses x close parentheses

Given that

  • fraction numerator straight d squared y over denominator straight d x squared end fraction space equals 30 x

  • fraction numerator d y over denominator d x end fraction equals 17 when x equals 1

find an expression for fraction numerator d y over denominator d x end fraction

How did you do?

10b
3 marks

Given that

  • y equals 40 when x equals 2

find the equation of the curve, y equals straight f open parentheses x close parentheses, giving your answer in simplest form.

How did you do?

11
6 marks

The figure below shows a sketch of the curve with equation y equals x left parenthesis x minus 1 right parenthesis left parenthesis x plus 2 right parenthesis.

q7-8-1-integration-hard-a-level-maths-pure-screenshot

Use algebraic integration to find the total area of the shaded regions.

How did you do?

1
Sme Calculator
5 marks

Use algebraic integration to find the value of

integral subscript 2 superscript 4 fraction numerator x cubed plus cube root of x over denominator 2 square root of x end fraction space d x

giving your answer correct to 3 significant figures.

How did you do?

2
5 marks

Find

integral open parentheses 2 minus x close parentheses cubed space straight d x

writing each term in simplest form.

How did you do?

3
5 marks

The function straight f open parentheses x close parentheses has the following properties

  • straight f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses equals 6 left parenthesis x minus 2 right parenthesis

  • straight f to the power of apostrophe open parentheses 2 close parentheses equals 8

  • straight f open parentheses 3 close parentheses equals 20

Find straight f open parentheses x close parentheses, giving your answer in simplest form.

How did you do?

4
6 marks

The figure below shows the straight line with equation 5 y equals 14 minus 3 x and the curve with equation 5 y equals 20 minus 2 x minus x squared.

The shaded region R is bounded by the curve and the straight line.

q9-8-1-integration-hard-a-level-maths-pure-screenshot

Use algebraic integration to find the exact area of R.

How did you do?

5
7 marks

The curve with equation y equals x cubed minus 2 x squared minus x plus 2 is shown in the figure below.

q8-8-1-integration-vh-a-level-maths-pure-screenshot

Use algebraic integration to find the total area of the shaded regions.

How did you do?

1
6 marks

Given that

integral open parentheses 2 open parentheses 3 minus 1 half x close parentheses close parentheses to the power of p space straight d x equals negative 1 fourth x to the power of 4 plus...

where p is a positive integer and negative 1 fourth x to the power of 4 is the term with the highest power of x, find fully

integral open parentheses 2 open parentheses 3 minus 1 half x close parentheses close parentheses to the power of p space straight d x

writing each term in simplest form.

How did you do?

2
Sme Calculator
5 marks

Find the value of the constant q such that

integral subscript q superscript 4 q end superscript 5 x square root of x space straight d x equals 15 066

How did you do?

3
6 marks

A function, straight f left parenthesis x right parenthesis

  • has a factor of open parentheses 2 x minus 1 close parentheses

  • has a factor of open parentheses 3 x plus 2 close parentheses

  • has a second derivative of straight f apostrophe apostrophe open parentheses x close parentheses equals 2 open parentheses 18 x minus 5 close parentheses

Find straight f left parenthesis x right parenthesis.

How did you do?

4
Sme Calculator
8 marks

The straight line with equation 4 y equals 4 x plus 17 and the curve with equation y equals 3 x squared minus 8 x minus 16 are shown in the figure below.

The region bounded by the straight line, the curve and the x-axis is shaded.

q6-8-1-integration-vhard-a-level-maths-pure-screenshot

Use algebraic integration to find the exact area of the shaded region.

How did you do?